# Interactive Real Analysis

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## 2.3. The Principle of Induction

### Examples 2.3.4(a):

Use induction to prove the following statements:
1. The sum of the first n positive integers is n (n+1) / 2.
2. If a, b > 0, then (a + b)n an + bn for any positive integer n.
1. The sum of the first n positive integers is n (n+1) / 2

To use induction, we have to proceed in fours steps.

Property Q(n):
1 + 2 + 3 + ... + n = n (n+1) / 2

Check Q(1):
1 = 1 * 2 / 2, which is true.

Assume Q(n) is true:
Assume that 1 + 2 + 3 + ... + n = n (n+1) / 2.

Check Q(n+1):
1 + 2 + 3 + ... + n + n + 1 =
= (1 + 2 + 3 + ... + n) + (n + 1) =
= n (n+1) / 2 + (n+1) =
= n (n+1) / 2 + (2n + 2) / 2 =
= (n+1)(n+2) / 2
Hence, Q(n+1) is true under the assumption that Q(n) is true.
But that finishes the induction proof.

2. If a, b > 0, then (a + b)n an + bn for any positive integer n.

Using an induction proof, we have to proceed in four steps:

Property Q(n):
(a + b)n an + bn if a, b > 0

Check Q(1):
(a + b) a + b, which is true.

Assume Q(n) is true:
Assume that (a + b)n an + bn

Check Q(n+1):
(a + b)n + 1 = (a + b)n (a + b) (an + bn) (a + b) =
= an + 1 + an b + a bn + bn + 1 an + 1 + bn + 1
because a, b > 0. Hence, Q(n+1) is true under the assumption that Q(n) is true.
But that finishes the induction proof.
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